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Применение интервальной оценки при использовании алгоритмов аппроксимации звеньев запаздывания в промышленных системах управления..pdf

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128
ТРУДЫ БГТУ. 2015. ? 6. Физико-математические науки и информатика. С. 128?131
??? 62.50
?. ?. ??????
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? ???????????? ???????? ??????????
?????????? ????????????? ?????? ????????? ????????? ????????????? ??????? ???????????? ? ????. ??????????????? ??????????? ?????????? ?????????? ????????????? ? ?????????????? ??????? ???????? ? ?????????, ?????? ? ????????????? ????????? ??????????. ???
????????? ??????????? ????????????? ?????? ? ?????????????????? ????????????? ? ???????? ?????? ?????????? ?????? ????????? ????????????? ???????? ??????????. ? ???????? ????????? ????????????? ????????? ?????????? ????, ???????, ???????.
??????????? ????????? ?????????? ????????????? ???????? ? ???????? ????- ? ??????????????? ????????? ?? ???????????? ????????. ??? ?????????? ????? ???? ??????? ???????????? ??????? ????????? ?????????? ???????? ??????????.
??????????? ?????? ???????? ?????? ????????? ????????????? ??? ?????? ?????????? ?
??????? ?????????? ????????????? ??????? ???????????? ??? ????????? ???????????? ????????? ???????? ??????????. ??? ??????? ???????? ?????????? ?????? ?????? ????????????? ?
????????? ??????? ??? ????????? ?????????? ???????? ??????? ? ?????????? ?????????? ???
????????????? ?? ???????????? ????????.
???????????? ???????? ?????? ???????? ????????????? ? ??????????? ?? ????????? ????????????? ??????? ??????????.
???????? ?????: ????????????, ?????????????, ?????? ??????????, ??????, ??????????????, ?????????, ???????, ?????????????.
?. V. Lapeto
Belarusian State Technological University
THE APPLICATION OF INTERVAL ESTIMATER USING THE APPROXIMATION
ALGORITHMS OF DELAYS IN INDUSTRIAL CONTROL SYSTEMS
The paper presents a comparative analysis of different options of approximation of delay units into the
series. The possibility of using a mathematical approximation algorithms similar to those model of industrial
control systems was considered. To compare the results of modeling systems with approximation delay and
the original systems the analyses of the characteristics of frequency-control objects has been made.
The alternatives of approximation are the Pade expansion and approximation of Taylor and Laguerre.
While considering mathematical models of industrial objects we took into consideration the
deflection characteristics of objects within a certain interval of the nominal value. This deviation is
caused by an error of measurement of tools of parameters of control object.
The result is a set of frequency characteristics for control systems with different variants of
approximation of the delay units at different dynamic properties of control objects. For each option, the
decomposition analysis of characteristics in the frequency domain with various combinations of object
dynamics and intervals deviation of its characteristics from the nominal value is given.
An algorithm for selecting options of approximation depending on the frequency characteristics of
the control object has been proposed.
Key words: delay, approximation, the control object model, characterization, structure, frequency,
persistence.
????????. ???????? ?????????????????
????????? ???????? ?????????? ? ????????????? ????? ??????? ???????? ????? ? ???????, ???????? ???????, ????????? ????????????, ???????? ??????????? ? ? ????????? ??????? ???????? ? ?????????? ????????? [1].
????????????? ?????????, ??????????? ?
???????? ?????????? ? ?????????????, ?????????????? ? ??????? ????????????????
????????? ? ????????????? ??????????.
????????? ? ?????????????? ??????? ????
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?????????? ?????? ?????????? ? ????????????? [2].
???????? ?????. ????? ?? ?????????
???????? ?????????? ???????? ???????? ????????????? ????? (???. 1) ?? ?????? ?????????? ? ??????.
Применение интервальной оценки при использовании алгоритмов аппроксимации звеньев запаздывания 129
???. 1. ??????????? ????? ??????? ??????????
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???? ?????????????? ?????? ??????? ?????????? ??????????? ????? ????????? ???????????? ?????? ?, ??? ???????? ?????, ?? ?????????? ? ???????? ???????. ? ?????????, ?
??????????? ??????????????? ????????? ??????????? ???????? ???????? (????????? ?????? ?????? ????????????, ????????? ?????????? ?????, ?????????? ????? ? ?. ?.).
????? ???????, ??? ????????? ??????????
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??? ?????????, ??????? ???????????? ????????? ?? ??????????????. ? ?????????? ? ??????? ???????, ??????????????? ???????? ????????????, ??????????? ?????????????? ????????? ???????? ?????????? ????????? ???????.
? ?????? ???????? ??????????, ??????????? ??
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??????????? ???????? ?? ???????????? ?????????????? ???????.
? ?????? ???????, ??????? ????????????
????? ???? ?????? ????? ?????????? ???????????? ? ????????? ??????-???????????? ?
????????? ??????? [3]. ????????? ????????
???????????? ??? ???????? ????????? ????????, ?? ??????????? ????????????? ????????
????? ??????????? ?? ??? ??????????-???????
????????? ?????????????? (????).
?? ????????? ????????? ?????????? ?????
??????? ???????????? ???????? ?????? ??
???????? ???????????? ??? ????????????? ????? ????. ???????????? ??????? ?????? ?????
? ????? ???? ???????? ????????? ???????:
n
W? ( p ) =
(n + k )!
? k !(n ? k )! (? ?p)n?k
k =1
n
(n + k )!
? k !(n ? k )! (?p)n?k
k =1
??? n ? ??????? ??????????.
,
(1)
?????????? ????????? ?????????????? ?????? ?????. ???????? ????? ??????????? ?? ????????, ???????? ????? ????????????? ? ??????????????, ?. ?. ?? 0 ?? 1 ??. ? ????????
??????? ??????? ???????? ????????????, ?????? 10 ?. ????? ???????? ???????? ???????????? ?????????? ??? ????????????? ????????? ?
?????? ???????, ?????????? ??????? ??????????????? ? ??????????????. ?????????? ????????????? ????? ???? 2?4 ???????.
???????? ? ????????? ??????? (p = j?), ?????????????? ????????? ????????????? ?????????? ??????? ??? ????????? ??????? ???????.
?? ???? ??????????-??????? ????????? ??????????????? ??? «Real» ????????????? ?????????????? ???, ? «Imag» ? ??????.
???? ????????????? 2?4 ???????? ??????????? ??????, ?????? ??? ?????????? (???. 2)
????? ???????, ??? ???? ?????????? 4-?? ??????? ?????? ????????? ???? ????? ???????
????????????, ? ?????????????, ?????????? ?????? ????????????? ????? ??????????? ????????.
?? ?????? ???????????? ???????? ?????????? ??????? ???????????? ??????????? ???
????????? ????? ?????? ??????? ??????????,
??????? ? ???? ??????? ??????????? ?????????????? ?????? ? ??????????????, ?????
???????, ??? ???????? ????????????.
Imag
0,525
0,52
0,515
0,51
0,505
0,5
n=2
n=3
n=4
??????
????????????
?0,86 ?0,855 ?0,85 ?0,845 ?0,84 ?0,835
Real
???. 2. ??????????-??????? ?????????
?????????????? ????????????? ????
?????????? ???? ??????? ? ??????????????, ? 10 ??? ???????, ??? ?????? ????????
????????????, ? ????????????? ????????,
?????? 1. ??????????? ???????? ???????????
??????? ?????????? ??? ???????????? ?????????, ?????????? ? ????????? ? ?????????????? ?????????? (????????, ????????? ??????????? ???????? ????????).
??? ????? ?? ???. 3, ?? ????? ????????
(?? 0,1 ?? 1,0 ??) ???? ??????? ????????????
? ??? ?????????? ?????????? ??????? ?????????, ?????? ??? ?????????? ??????? ??????????
??????????. ?????????? ?????????? ????????????? 4-?? ???????.
130
А. В. Лапето
Imag
0,03
n=2
n=3
n=4
??????
????????????
0,02
0,01
Imag
1
0,5
0
0
?0,01
?0,5
?0,05
0
0,05 Real
?1
???. 3. ???? ????????????? ????
? ??????????? ????????
?1
?????????? ????????????? ????? ???????
???????????? ????? ??????? (????????? ?????), ??????????? ? ????? ???? ?????????
??????????:
n
1
( ? ?p ) k .
k
!
k =1
W? ( p ) = ?
(2)
???????????? ???? ?????? ? ????????
?????????? ????????? ??????? (???. 4).
????????? ???
??????
????????????
1,5
0,5
0,4
0,5
0,3
??? ??????????
?????????? 5%
?????????? 1%
???? 2-?? ???????
0,2
?0,87 ?0,86 ?0,85 ?0,84 ?0,83
?0,5
?1
1 Real
?? ?????????? ???? ???????, ??? ??? ?????????? ??????? ???????? ???????? ????? ?????? ??? ??????? ???????????? ? ??? ??????????????? ???????????.
?????????? ?????????? ????????????? ????
? ????????? ? ??????????? ????????, ??????????? ???????????? ???????? 2-?? ??????? (???. 6).
1
0
0,5
???. 5. ???? ??????????? ???????
Imag
Imag
n=2
n=3
n=4
??????
????????????
?0,5
0
?0,5
0
0,5
???. 6. ???? ? ??????????? ????????
????????????
Real
???. 4. ???? ????????????? ????????? ?????
?? ?????? ???????? (0?0,2 ??) ?????? ????????????? ? ??????? ???????????? ?????????, ?????? ??? ?????????? ??????? ?????????
???????????? ??????????????, ? ????????????????? ?????????????? ?????????? ?????
?????????? ? ?????????????.
????? ????????????? ????????? ??????????, ?? ???????? ????? ???????????? ????????????? ???????????? ???????:
W? =
(1 ? p / 2n) n
.
(1 + p / 2n) n
Real
(3)
?????? ????????? ????????????? ? ????????? ??????? ??? ??????-????????????
??????? 2?4 ??????? (???. 5).
?? ???????? ??? ???????????? ???????????? ??????? ??????? ??????????, ??? ? ???????? ???????????? ????? ??????????? ?? ??????
???????????? ????????. ??? ????? ???? ??????? ?????????? ?????????, ?????? ??? ????????? ??????-?????????? ??????? ?????,
??????????, ????????????? ?????????????
????????? ? ?. ?.
?? ???. 6 ???????????? ?????? ???? ???
??????? ????????????, ?????????? ????, ? ?????
??? ?????????? ???????? ???????????? ? ????????? 1 ? 5% ?? ???????????? ????????.
????? ????????, ??? ????????????? ????????????? ?????? ?????????? ??????, ??? ????????? ?????? ??????? ?????????? ? ????????
??????. ????? ????????? ??????? ??????????
?????????? ??? ?????? ??????. ?? ???????
Применение интервальной оценки при использовании алгоритмов аппроксимации звеньев запаздывания 131
???????? ??????? ????????????? ??????????
???????????? ??????? ?????????? ?????????????? ???????.
??????????. ?????????? ???? ???? ????????? ???? ?? ????????? ? ?????????? ?????????? ?????????????. ? ?????? ??????? ? ???????
?????????????, ??????????? ??????????? ???????? ????????????, ????? ???????? ??????
???????? ??????????. ????????? ???????????
???????? ?????????? ? ?????????????? ??????????????? ???????? ??????? ? ?? ???????????
??????? ?????????? ??????, ?? ??? ????? ???????
??????????? ????? ???????????? ???????? ??????????, ?? ??????????? ???????.
?????????? ???????????? ? ????????? ???
???????? ???????????? ?????? ?? ?????????
?????? ????????, ??? ??? ??? ?????? ????? ???????????. ????? ??? ?????? ???????? ???????
????? ?????????? ????????? ? ???? ??????????
??????????????? ????????????.
??????????? ??????? ??????????????? ??????? ??????????? ?? ????, ??? ? ??????????
????, ?????? ??????? ????????? ??? ? ??????? ?? ?????????? ???? ???? ??????????? ???????????? ????? ??????? ????????????? ??
????? ??????? ?????? ??????. ????????? ????? ??????-???????????? ??????? ?????,
??? ? ???? ????, ? ?????????????, ? ?????, ????????????? ?? ????????? ?????????? ??????????, ??????. ????????? ??????? ????????????? ????? ????????????, ? ???????, ???
??????????? ???????? ?? ?????? ??????? ?
???????????? ???????? ? ?????? ? ???????????? ?????????, ???????? ??????????? ??????? ????????.
??? ????????? ???????? ???????????? ??
???????????? ???????? ?????????? ? ??????????????? ??????? ?????????? ???????? ??????? ????????, ??? ??????? ?????????????
???? ????????? ?????????? ???????. ?????? ????? ????????, ??? ????? ??????????
??????????? ?????? ?? ?????? ????????, ??????????? ??? ???????????? ????????
??????????.
??????????
1. ?????????? ?. ?. ?????????? ????????? ? ?????????????. ?.: ?????, 1978. 416 ?.
2. ?????? ?. ?. ?????? ??????? ??????? ?????? ??????????????? ?????????? ? ????????????? //
????? ????. 2011. ? 6: ???.-???. ????? ? ???????????. ?. 78?80.
3. ?????? ??., ??????-?????? ?. ????????????? ????. ?.: ???, 1986. 502 ?.
References
1. Yanushevskiy R. T. Upravlenye ob'ektami s zapazdyvaniem [Managing objects with delay].
Moscow, Nauka Publ., 1978. 416 p.
2. Lapeto A. V. Analysis of methods for the synthesis of automatic control systems with delay. Trudy BGTU
[Proceedings of BSTU], 2011, no. 6: Physical-mathematical sciences and informatics, pp. 78?80 (In Russian).
3. Beyker Dzh., Greyvs-Morris P. Approksimatsii Pade [Pade Approximants]. Moscow, Mir Publ.,
1986. 502 p.
?????????? ?? ??????
?????? ????????? ?????????? ? ????????? ??????? ????????????? ???????????????? ????????? ? ??????????????. ??????????? ??????????????? ??????????????? ??????????? (220006,
?. ?????, ??. ?????????, 13?, ?????????? ????????). E-mail: AVLapeto@gmail.com
Information about the author
Lapeto Aleksandr Vasil?evich ? assistant, the Department of Automation of Production Processes and
Electrical Engineering. Belarusian State Technological University (13a, Sverdlova str., 220006, Minsk,
Republic of Belarus). E-mail: AVLapeto@gmail.com
????????? 22.03.2015
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